Dynamics of an epidemic model with impact of toxins

Abstract

We have considered a toxin-dependent dynamical model to study the effects of an environmental toxin on the spread of infectious diseases in the population. The well-posedness of the model is discussed by showing the positivity and boundedness of the state variables. Model analysis is performed as well as the stability and bifurcation of equilibrium points are also established. Numerical simulation is performed to verify the theoretical results and it shows that the system possesses a transcritical bifurcation around the disease-free equilibrium. Our analytical results indicate that there exists a threshold value of the environmental toxin, i.e., if the environmental toxin amount is lower than the threshold, the system has a stable disease-free equilibrium and if the environmental toxin amount is higher than the threshold value, then the system has a unique endemic equilibrium. Numerical simulation also shows that the environmental toxin plays a crucial role in spreading of infectious diseases. An optimal control problem has been formulated in the later part to minimize the cost and disease fatality by choosing the treatment and the depuration of the toxin as control variables. Numerical analysis indicates that if only control via depuration is used, then it will be more economical for the time period whereas treatment works well but with a lesser intensity. Moreover, simultaneous use of both the control interventions is more useful for the system dynamics and it reduces the number of infective individuals and also minimizes the economic cost generated from disease burden. This study gives a clear view of the impact of toxins on the spread of infectious diseases in the population and it helps the disease control agencies and governments to take suitable precautions to control the disease.